Carrier frequency parameter

In an expansion $\sum_m a_m e^{i\kappa m\Phi}$ , the chosen positive number $\kappa$ is a carrier frequency parameter. The integer $m$ is the harmonic multiplier; $\kappa m\Phi$ is the full phase. This convention separates the common rapid oscillation scale from its harmonics.

Periodicity and physical scale

If $\Phi=p\theta+\Phi_0$ with $\theta$ of period $2\pi$ and $\Phi_0$ independent of $\theta$ , then $e^{i\kappa\Phi}$ is angularly periodic precisely when $\kappa p\in\mathbb Z$ . The physical wavevector of harmonic $m$ is $\kappa m\nabla\Phi$ . A carrier parameter, a dyadic localization scale, and the physical wavenumber serve different roles.